Extremal Graphs for Blow-Ups of Keyrings

Abstract

The blow-up of a graph H is the graph obtained from replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. Given a graph H and a positive integer n, the extremal number, ex(n, H), is the maximum number of edges in a graph on n vertices that does not contain H as a subgraph. A keyring $C_s(k)$ is a $(k+s)$-edge graph obtained from a cycle of length k by appending s leaves to one of its vertices. This paper determines the extremal number and finds the extremal graphs for the blow-ups of keyrings $C_s(k)$ ($k\ge 3$, $s\ge 1$) when n is sufficiently large. For special cases when $k=0$ or $s=0$, the extremal number of the blow-ups of the graph $C_s(0)$ (a star) has been determined by Erdös et al. (J Comb Theory Ser B 64:89–100, 1995) and Chen et al. (J Comb Theory Ser B 89: 159–171, 2003), while the extremal number and extremal graphs for the blow-ups of the graph $C_0(k)$ (a cycle) when n is sufficiently large has been determined by Liu (Electron J Combin 20: P65, 2013).